There are many excellent, fast algorithms for tessellating spaces of
any dimension. We used the Matlab routine Qhull [1]} which is
computationally efficient and returns adjacency and other auxiliary information in
a convenient form. Without any further computations, the Voronoi cells
express considerable statistical information about the point distribution.
For example, Figure REF  shows the distribution
functions of the local densities computed as the reciprocal of the
volumes of the Voronoi cells for the three cases: the SDSS DR7 data,
the Millennium Simulation data, and the uniform data. These distributions
characterize the dynamic range of the cell sizes.
As expected, the cells in the uniform case have
a relatively narrow distribution centered around the mean cell size,
while in the other cases a broader range reflects the presence of
structure on a wider range of scales.
The degree to which the distribution for the case of the MS
data is similar to that for the DR7 data confirms the correctness of
this aspect of the simulations. While the log densities are approximately
normally distributed, the density distributions themselves have long
tails that render the (log) of the mean value a misleading central measure.
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